For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.*

Warm-Up

30 minutes

Students come into class and get started directly on a word problem on the Warm-Up—the goal is for them to figure out their own tools for tackling the problem. If students struggle to start, don’t tell them how to start. Ask them how to start: “How could we start? What would be a good way to understand this problem better?” Even if it takes three times longer for them to get going this way, you send them the message that they when they struggle they will need to figure things out. If you just tell them, “Make a data table,” then they will just keep asking you every time. Also, I always have lots of graph organizers available and already photo-copied so that I can give them to students who need more scaffolding can ask for it.

I tell students that at this point, I want them to figure out how to solve problem (2) even if they aren’t sure. The two data tables are provided as a small hint to students about how to look at the problem, but most of their confusion will happen at the endpoint of . Again, rather than tell them about how to deal with this, ask them: “What is going on at ?What do those symbols even mean? How should the graph look?” Even if they don’t come up with the right answers, they will be thinking, which is the goal (MP1). I ask students to show their different ideas and see if anybody agrees or disagrees with them (MP3). I do this whether anybody is right or not, but usually through this process the correct understanding surfaces.

Problem (3) is more of a challenge, so I ask them to figure out as much as they can about it and I tell them that they have a few more days to think about this. The purpose of doing this is to communicate the message to students that they don’t need to wait for me to tell them about how to do math, but that they can think about it and figure it out on their own. It is also great for differentiation, because advanced students have the chance to construct knowledge with few scaffolds, while less advanced students get exposed to it and get more scaffolds in upcoming lessons.

Compare and Contrast Piecewise Functions Warm-Up.docx

Investigation and New Learning

Closing

10 minutes

Again, the questions for the closing bring the lesson to a higher level of thinking. Students might have no idea about how to answer these questions when you first ask them, but with 10 minutes and some discussion with a partner and sharing of ideas, students can gain some understanding. The big idea for the question (A) is that it doesn’t matter which of the endpoint inequalities is strict if the function is continuous because the outputs will be the same either way, whereas it does matter when the function is not continuous. Students may use the phrase “match up.” When the end points “match up,” it doesn’t matter. This is fine for now.

The second question is pushing student understanding: it is so easy to identify continuity by looking at the graph, how can you do so by looking at the function rule alone? At this point, I ask: “Where might continuity break down? Where could we have a problem with continuity?” The goal is to get them talking about the endpoints.

The whole purpose of these kinds of conversations and questions is to focus the conversation on the aspect of the learning unit that is most challenging for students and to bring up all the misconceptions. Even if everything is not resolved in these conversations, the misconceptions are highlighted and the more confusing aspects of the unit come to the surface.